It might indeed be argued, with some show of probability, that the condition of successive consciousness is essentially the condition of a finite and imperfect intelligence, consequent only upon its very limited power of simultaneous consciousness. The scholastic doctrine of an eternal Now, or nunc stans, so contemptuously treated by Hobbes, in this respect contains assuredly no prima facie absurdity. The error of such speculations is of another kind. It consists in mistaking the negation of all thought for an act of positive thinking. As our whole personal consciousness is subject to the condition of successiveness, we can form no positive notion of a different state we only know that it is something which we have never experienced. The nature and attributes of an Infinite Intelligence must be revealed to us in a manner accommodated to finite capacities. How far the accommodation extends, we have no means of determining, as we cannot examine the same data with a different set of faculties. The importance of this distinction between positive

1 Vide Boeth. De Consol. Phil., lib. v. pros. vi.

2 It is surprising to see how near some of the earlier views on this point approached to, without actually arriving at, the doctrine of Kant. Had the question been considered subjectively as well as objectively, on the psychological as well as on the metaphysical side, the most important conclusion of the Critical Philosophy would have been anticipated. When Hobbes, in his controversy with Bramhall, said, "I never could conceive an ever-abiding now," he was right; but he was wrong in supposing that this was decisive of the point at issue. We can only conceive in thought what we have experienced in presentation; and all our past presentations have been given under the law of succession. But this does not enable us to decide what may be the condition of other than human intelligences. In this respect, the remark of Bramhall is exactly to the purpose: "Though we are not able to comprehend perfectly what God is, yet we are able to comprehend what God is not; that is, he is not imperfect, and therefore he is not finite." Reid (Intell. Powers, Essay iii. ch. 3) treats the nunc stans as a contradiction, which it is not.

and negative thinking will be more closely examined hereafter.

But, to return to the question of mathematical necessity: To construct the whole science of Arithmetic, it is only requisite that we should be conscious of a succession in time, and should be able to give names to the several members of the series. And since in every act of consciousness we are subject to the law of succession, it is impossible in any form of consciousness to represent to ourselves the facts of Arithmetic as other than they are. To the art, not to the science, of Arithmetic belong all the methods for facilitating calculation which imply anything more than the mere idea of succession. Such a method, and a powerful one, is afforded by the invention of Scales of Notation, in which, to the idea of succession, is added that of recurrence; the series being regarded as commencing again from a second unit, after proceeding continuously through a certain number of members, ten for example, as in the common system. Hence we are enabled to repeat over again, in the second and subsequent decades, the operations originally performed in the first, and thus indefinitely to extend our calculus in the form of a continually recurring series; but the calculus, though thus rendered infinitely more efficacious as an instrument, remains in its psychological basis unaltered.

From these considerations it follows that the several members of an arithmetical series are incapable of definition. Succession in time, and the consciousness of one, two, three, etc., are not complex notions abstracted from and after a multitude of intuitions, but simple immediate intuitions, differing, as far as numeration is concerned, only in the order of their presentation. They are not by any act of thought compounded, the latter from the earlier:

they cannot be resolved into any simpler elements of consciousness, presentative or representative, being themselves the à priori conditions of consciousness in general. Hence the failure of all attempts to analyze numerical calculation as a deductive process. Leibnitz, and subsequently Hegel, have endeavored to represent the arithmetical processes as operations of pure analysis. Assuming, for example, 12 and 7 and 5, as given concepts, they show that the first may be ultimately analyzed into the same constituent units as the two last; and this is regarded as an explanation of the whole process of Addition. They overlook the fact that, in that process, 12 is not given, but has to be determined by the addition of the two other numbers. Arithmetic is not, like Geometry, a science whose definitions are genetic and preliminary to its processes. The analysis of any number into its constituent units presupposes the whole operation which it professes to give rise to. We may call, if we please, such an analysis definition; but we must not suppose that it in any degree corresponds to the definitions of Geometry, or answers the same purpose in the operations of the science.1

The above considerations are sufficient for our present purpose, which is to determine the psychological basis of mathematical judgments, and their consequent special character as necessary truths, in a distinct sense from that in

1 Writers of a very different school from that of Leibnitz or Hegel have fallen into a similar error with regard to the nature of arithmetical processes. Mr. Mill, for example, regards the whole science of numbers as derived from the common axioms concerning equality, and the definitions of the several numbers. Stewart appears to have been of the same opinion. On the contrary, the whole essentials of the science must be in existence before the so-called definitions can be formed. The applications of the calculus as an instrument must not be confounded with its essential constituents as a science.

which the term is applied to logical or physical principles. Mathematical judgments are synthetical, based on the universal conditions of our intuitive faculties, and are necessary, not, properly speaking, as laws of thought, but because thought can only operate in conjunction with matter given by intuition, and intuition cannot be emancipated from its own subjective conditions. Hence we are compelled to think of our intuitions under the same laws according to which they are invariably realized in consciousness. Judgments of logical necessity, on the other hand, are analytical, and rest on the laws of thought, properly so called. Their analytical character is a necessary consequence of the constitution of the thinking faculty, and is so far from being a proof of the unsoundness or frivolity of logical speculations, that it is the strongest evidence of their truth and scientific value, and leads to most important consequences, both in Logic and in Psychology.

The nature of these judgments, as well as of those distinguished as metaphysically necessary, will be examined in the following chapters.

CHAPTER V.

ON THE PSYCHOLOGICAL CHARACTER OF METAPHYSICAL

NECESSITY.

A DISTINCTION between necessary and contingent matter is found, somewhat out of place it is true, but still it is found, in most of the older, and, among English writers, in most also of the recent treatises on Logic.1 The boundaries of each, however, are not in the majority of instances determined with any approach to accuracy. Among the schoolmen, the favorite example of a proposition of the highest degree of necessity was omne animal rationale est risibile; an example consistent enough with the mediaval state of physical science, but which in the present day will scarcely be allowed a higher degree of certainty than belongs to any other observed fact in the constitution of things. An eminent modern Logician gives as an example of a proposition in necessary matter, "All islands are surrounded by water;" an example which is only valid in

1 Matter in this sense must not be confounded with the modality recog nized by Aristotle, and by most of the modern German Logicians. The former is an understood relation between the terms of a proposition, — the form of the proposition being in all cases "A is B," and is supposed to be of use in determining the quantity of indefinites. The latter is an expressed relation, the form of the necessary proposition being "A must be B;" and this is applicable to universal and particular propositions indifferently. The admission of the latter is still a point of dispute among eminent authorities; the admission of the former will be tolerated by no Logician who understands the nature of his own science.